Quasilinear Schrödinger-Poisson system under an exponential critical nonlinearity: existence and asymptotic of solutions

Abstract

In this paper we consider the following quasilinear Schrödinger-Poisson system in a bounded domain in ${\mathbb{R}}^2$: $$ \left\{ \begin{array}[c]{ll} - Δu +ϕu = f(u) &\ \mbox{in } Ω, -Δϕ- \varepsilon^{4}Δ_4 ϕ= u^{2} & \ \mbox{in } Ω, u=ϕ=0 & \ \mbox{on } \partialΩ\end{array} \right. $$ depending on the parameter $\epsilon >0$. The nonlinearity $f$ is assumed to have critical exponencial growth. We first prove existence of nontrivial solutions $(u_{\epsilon },{\varphi }_{\epsilon })$ and then we show that as $\epsilon \to 0^{+}$ these solutions converges to a nontrivial solution of the associated Schrödinger-Poisson system, that is by making $\epsilon =0$ in the system above.

References in corpus (1)

• Existence and asymptotic behaviour of solutions for a quasi-linear schrodinger-poisson system under a critical nonlinearity